On a Gdel's misjudgment

Giuseppe Ragun

UCAM - Universidad Catlica de Murcia, Spain -

graguni@ucam.edu

Abstract

The fundamental aim of the paper is to correct an harmful way to in-

terpret a Gdel's erroneous remark at the Congress of Knigsberg in 1930.

Despite the Gdel's fault is rather venial, its misreading has produced and

continues to produce dangerous fruits, as to apply the incompleteness

Theorems to the full second-order Arithmetic and to deduce the semantic

incompleteness of its language by these same Theorems. The rst three

paragraphs are introductory and serve to dene the languages inherently

semantic and its properties, to discuss the consequences of the expression

order used in a language and some question about the semantic complete-

ness: in particular is highlighted the fact that a non-formal theory may be

semantically complete despite using a language semantically incomplete.

Finally, an alternative interpretation of the Gdel's unfortunate comment

is proposed.

KEYWORDS: semantic completeness, syntactic incompleteness, cate-

goricity, arithmetic, second-order languages, paradoxes

1 Formal systems

Often the adjective formal is abused in violation of the original meaning that

can rightly be called Hilbertian. Actually, it is common understanding formal

system as a synonym for axiomatic system, although no one doubts the need

for non-formal axioms in Mathematics. This superciality, together with the

wrong meaning given to certain armations of Gdel, produces the serious

mistakes that we will show.

The best denition of a formal system is probably that one given by Lewis

in 1918

1

:

A mathematical system is any set of strings of recognizable marks

in which some of the strings are taken initially and the remainder

derived from these by operations performed according to rules which

are independent of any meaning assigned to the marks.

1

C. I. Lewis, A Survey of Formal Logic, Berkeley University of California Press (1918), p.

355.

1

So, a framework in which, starting by certain strings of characters with no

meaning (the formal axioms), are produced (deducted) other strings with no

meaning (the theorems) by use of operations (deductive rules) which never make

use of any meaning assigned to any mark. To get from that sort of abacus a

real scientic discipline, we must interpret - if this is possible - the characters

and strings so that: a) the axioms are true; b) the deductive rules are sound, ie

capable of producing only true meaningful strings, starting by true meaningful

strings. Such an interpretation is said sound model. This conceptualization,

revolutionary for the times, is also present in the contemporary Bernays and

Post

2

as well - perhaps for the very rst time - in Hilbert, as it can be noted in

many passages of his correspondence with Frege (1899-1900):

[...] It is surely obvious that every theory is only a scaolding

or schema of concepts together with their necessary relations to one

another, and that the basic elements can be thought of in any way

one likes. If in speaking of my points I think of some system of things,

e.g. the system: love, law, chimney-sweep . . . and then assume all

my axioms as relations between these things, then my propositions,

e.g. Pythagoras' theorem, are also valid for these things. In other

words: any theory can always be applied to innitely many systems

of basic elements.

[...] a concept can be xed logically only by its relations to other

concepts. These relations, formulated in certain statements I call

axioms, thus arriving at the view that axioms . . . are the denitions

of the concepts. I did not think up this view because I had nothing

better to do, but I found myself forced into it by the requirements

of strictness in logical inference and in the logical construction of a

theory

3

.

Incidentally, the fact that none of the mentioned authors uses rigorously the

adjective formal is not surprising: since for all of them the forthright belief (or

hope) was that every axiomatic theory was formalizable, the formal dressing

was nothing more than the correct way of presenting any axiomatic discipline.

Precisely Gdel is one of the rst to use the term carefully and accurately for

his attention to distinguish meticulously between Mathematics and Metamath-

ematics and his consciousness of expressive limits for the formality (those that

he self will help to shape). As a matter of fact, even without resorting to the

feasibility of the condition of categoricity (which only applies in non-formal ax-

iomatic systems

4

), the non-formal axiomatic theories are necessary to express,

and possibly decide, all that the Mathematics was thought to deal.

The concept of formality requires at least a clarication. The question is

simply whether the formal systems coincide with recursively (or eectively, as-

2

See, for example, the Dreben and Heijenoort introduction to rst Gdel's works in Kurt

Gdel collected works, ed. Feferman et al., Oxford univ. press (1986), vol. 1, pp. 44-48.

3

From two Hilbert's letters to Frege, in: G. Frege, Philosophical and Mathematical Corre-

spondence, ed. G. Gabriel, et al. Oxford: Blackwell Publishers (1980).

4

If we limit ourselves to the consideration of theories with models of innite cardinality.

2

suming the Church-Turing Thesis) axiomatizable systems. When the works of

Church and, especially, Turing

5

convinced Gdel that the machine models con-

sidered by them were, in fact, completely equivalent to his general recursive

functions (ie convinced him of the Church-Turing Thesis), he, in a footnote

added in 1963 to the text of his most famous work, wrote:

In consequence of later advances, in particular of the fact that

due to A. M. Turing's work a precise and unquestionably adequate

denition of the general notion of formal system can now be given,

a completely general version of Theorems VI and XI [the two in-

completeness Theorems] is now possible. That is, it can be proved

rigorously that in every consistent formal system that contains a

certain amount of nitary number theory there exist undecidable

arithmetic propositions and that, moreover, the consistency of any

such system cannot be proved in the system

6

.

clarifying, in a footnote of this same text, that:

In my opinion the term formal system or formalism should

never be used for anything but this notion. [...]

The Gdel's proposal was therefore to coincide, by denition, the eectively

axiomatizable systems with the formal systems. But this suggestion was not

successful. Certainly, it is just a matter of convention; but if we maintain

the original denition of formal, the two concepts are dierent. Consider a

theory that deduces only by the classical deductive rules. Since these rules are

mechanizable, the only obstacle for a machine, in simulating the system, is the

specication of the set of axioms: if it is mechanically reproducible, the system

will be eectively axiomatizable. On the other hand, on the basis of denition

of formality, the chains to call axioms must be exhibitable but can be specied

in arbitrary way. So, eective axiomatizability implies formality but not vice

versa.

An important example of this case can be built by the formal Peano Arith-

metic (PA). Assuming that such theory is consistent, we can add as new axioms

the class of its true statements in the intuitive (or standard) model. By con-

struction, it forms a syntactically complete system (PAT ) which, as a result of

the rst incompleteness Theorem, cannot be eectively axiomatizable. Never-

theless this system can still be called formal : it is necessary to give meaning

to the sentences just to determine whether or not they are axioms. Once done,

every statement can be reconverted to a meaningless string, since the system

can deduce its theorems without making use of signicance. So, really exists a

formal system able to solve the halting problem: cold comfort.

5

A. M. Turing, On computable numbers, with an application to the Entscheidungsproblem,

Proceedings of the London Mathematical Society (2) 42 (1937), pp. 230-265.

6

K. Gdel, On formally undecidable propositions of Principia mathematica and related

system I, in Kurt Gdel collected works, op. cit., vol. 1 p. 195.

3

2 Semantic languages and non-formal systems

Consider an arbitrary language that, as normally, makes use of a countable

7

number of characters. Combining these characters in certain ways, are formed

some fundamental strings that we call terms of the language: those collected in

a dictionary. When the terms are semantically interpreted, ie a certain meaning

is assigned to them, we have their distinction in adjectives, nouns, verbs, etc.

Then, a proper grammar establishes the rules of formation of sentences. While

the terms are nite, the combinations of grammatically allowed terms form an

innite-countable amount of possible sentences.

In a non-trivial language, the meaning associated to each term, and thus to

each expression that contains it, is not always unique. The same sentence can

enunciate dierent things, so representing dierent propositions. For example,

the same sentence it is a plain sailing has a dierent meaning depending on

the circumstances: at board of a ship or in the various cases with gurative

sense. How many meanings can be associated to the same term? That is: how

many dierent propositions, in general, can we get by a single sentence? The

answer, for a normal semantic language, may be amazing.

Suppose we assign to each term a nite number of well-dened meanings. We

could then instruct a computer to consider all the possibilities of interpretation

of each term. The computer, to simplify, may assign all the dierent meanings

to an equal number of distinct new terms that it has previously dened. For

example, it might dene the term sailing-f for the gurative use of sailing.

The machine would then be able, using the grammar rules, to generate all

the innite-countable propositions. In this case we will say that, in the specic

language, the meaning has been deleted

8

. More generally we have this case when

the dierent meanings allowed for each term are eectively enumerable: even in

the case of an innite-countable amount of meanings, the computer can dene

an innite-countable number of new terms and associate only one meaning to

each term in order to establish a biunivocal correspondence between sentences

and propositions. So, the machine could list all them by combination.

Hence, by denition, we will say that a language is inherently semantic (ie

with a non-eliminable meaning) if it uses at least one term with an amount not

eectively enumerable of meanings; with the possibility, which we will comment

soon, that this quantity is even uncountable. From the fact that a sentence

represents more than one proposition if and only if it contains at least one

term dierently interpreted, it follows an equivalent condition for the inherent

semanticity: a language is inherently semantic if and only if the set of all possible

propositions is not eectively enumerable.

A rst important example of such a language was considered in the previous

section. The fact that the axioms of PAT are not eectively enumerable proves

7

Finite, as the usual alpha-numeric symbols, or, to generalize, innite-countable. In this

paper we will use either countable or enumerable with the same meaning, ie to indicate

that there exists a biunivocal correspondence between the considered set and the set of the

natural numbers.

8

Just a concise choice rather than the more correct mechanically reproduced.

4

that in the expression true statement in the standard model the term true

has got an amount not eectively enumerable (although enumerable) of distinct

meanings. So, the phrase belongs to an inherently semantic language.

When the set of all possible propositions is enumerable but not eectively

enumerable, still it is possible to dene an amount innite-countable of new

terms and to associate only one meaning to each term (so re-establishing a biu-

nivocal correspondence between sentences and propositions); but this operation

cannot be performed by a machine. Go back to the example of PAT : an inher-

ently semantic phrase is used to dene all its new axioms, but then every axiom

is formulated by its unique symbolic representation in PA. Every proposition of

PAT corresponding to the expression true statement in the standard model is

explicitly replaced with an appropriate formula of PA. So, in this case, the def-

inition of new terms consists precisely in using the formulas of PA to express

the propositions of PAT.

Now consider the case of an inherently semantic language in which all pos-

sible propositions are not enumerable, ie in which there exists at least one term

with an uncountable quantity of meanings. Due to the uncountability of prop-

erties of the standard natural numbers (N ), ie of the set of all subsets of N,

P(N), this case is especially interesting, because only a language of this kind is

capable of expressing these properties. This time it is not possible to dene new

terms so to associate to them an unique meaning, because the number of all

possible strings is only countable

9

. Not even we can reaccommodate the things

so to re-associate a quantity at most innite-countable of interpretations to each

term, because in this way we still could get only a countable total amount of sen-

tences. There is no way to avoid that at least one term conserves an uncountable

number of interpretations.

At rst, this feature might surprise or even be considered unacceptable;

but it's really what happens in every usual natural language. Of course, we

should not assume that all the meanings that we ever can assign to the term

deal, for example, have to be specied once and for all (that is obviously

impossible). Furthermore, there is no doubt that all the meanings that ever

will be assigned to this word are only a countable number (indeed: nite!).

But the fact remains that the possible interpretations of the term vary inside an

innite collection; moreover, a collection not limited by any prexed cardinality.

Some classic paradoxes can be interpreted as a conrmation of this property.

The Richard's one

10

, for example, can be interpreted as a meta-proof that the

semantic denitions are not countable, ie that they are conceivably able to dene

each element of a set with cardinality greater than the enumerable one (and

therefore each real number). The proper technique used in diagonal reasoning,

9

Sometimes the uncountability is interpreted simply as a non-formalizable countability,

ie it is achieved by an inherently metamathematical connection. Even according to this view,

however, this countability cannot be capable to assign a dierent symbolic string to each

element of the uncountable set: otherwise, nothing could prohibit to formalize this function

inside the formal Set Theory.

10

Where rstly it is admitted that all possible semantic denitions of the real numbers are

in a countable array and, then, one can dene, by a diagonal criterion, a real number that is

not present in the array.

5

reveals that the natural language is able to adopt dierent semantic levels (or

contexts) looking from the outside what was before dened, namely, what

was previously said by the same language. Identical words used in dierent

contexts have a dierent meaning and for the number of contexts, including

nested, there is no limit. On the other hand, the Berry's paradox clearly shows

that a nite amount of symbols, dierently interpreted, is able to dene an

innite amount of objects. Here the key of the argument is again the use of two

dierent contexts to interpret the verb dene.

Finally, we consider a consistent arbitrary axiomatic system (S

A

). We wish

that S

A

is able to express and possibly decide

11

all the properties of the natural

numbers. In particular it must be capable to distinguish the properties one from

the other, in order to deduce, in general, dierent theorems starting by dierent

properties. Admitting, as normally, that S

A

makes use of a countable number

of symbols, if we interpret the theory in a certain conventional model, this

will associate a single meaning to each sentence. So, the interpreted sentences

will be only an enumerable amount and they cannot express all the properties

of the natural numbers. Therefore, the only possibility is to consider a non-

conventional model of the system, able to assign more than a single meaning

(indeed a quantity at least 2

0) to at least one sentence; and, then, able to

verify the remaining requirements for a normal model

12

. That is what really

happens in the so-called full second order Arithmetic (FSOA): its standard

model, which can be proved unique under isomorphism, is of this kind. The

result, of course, is a non-formal axiomatic system. In fact, as a consequence

of what we have observed, dierent properties with identical formulas must

exist; then, admitting the formality, one could deduct by them always identical

theorems, in disagreement to our demands. Therefore, no kind of interpretation

can allow an axiomatic system to express and study all the properties of the

natural numbers in compliance with formality.

3 Questions about the expression orders

The rst-order predicate calculus of Logic

13

, or briey rst-order classical Logic,

is a formal system which constitutes the structural core (ie the language) of the

ordinary axiomatic theories. In it, the existential quantiers and 14 only canrange over variable-elements of the universe (U ) of the model. In 1929 Gdel

proved the semantic completeness of this theory, namely, that all the valid (ie

11

By decide a sentence, we mean to conclude that it or its negation is a theorem.

12

As a result, this model cannot be confused with a conventional model of a formal (and

maintained formal) system, equipped with an uncountable universe. This one, considered for

example for PA, will continue to assign only one meaning to each sentence of the system.

And, on the other hand, an uncountable amount of exceeding elements of the universe will

have no representation in PA.

13

Dened by Russell and Whitehead in Principia Mathematica, Cambridge University Press

(1913).

14

Really, only one of the two is strictly necessary.

6

true in every model) sentences are theorems

15

. The widest generalization of

the Gdel's result is due to Henkin

16

and extends the semantic completeness to

every formal system.

The second-order predicate calculus of Logic, or second-order classical Logic,

extends the use of existential quantiers to predicates, ie to properties of the

elements of U. From the standpoint of the Set theory, the predicate-variables

vary inside the set of all subsets of U, that is, P(U). In non-trivial cases, U

is innite and consequently P(U) always has got an uncountable cardinality.

If we consider an arbitrary non-trivial axiomatic theory based on second-order

classical Logic, we have thus three fundamental cases:

1. The axioms and/or rules of the theory nevertheless ensure formality. For

this purpose it is necessary (although not sucient) that they limit the

variability of the predicates within a countable subset of P(U). In the typ-

ical case, this is achieved by means of appropriate comprehension axioms.

This case is known as general, or Henkin's, Semantics.

2. There is no limit for variability of the predicates in P(U). This full un-

derstanding is known as standard Semantics. In any case a non-formal

system is obtained, since it is possible to express an uncountable quantity

of sentences.

3. The variability of predicates in P(U) is restricted but not enough to verify

the formality. The sentences may be uncountable or countable but, in the

latter case, not eectively.

In case 1) the theory always is semantically complete; what happens in the

remaining cases?

The FSOA Arithmetic is right an important example of case 2). In it, an

induction principle, valid for any property of the natural numbers, is dened as

axiomatic scheme. Denitely, this is a full understanding of the inductive rule,

that implies uncountability for the sentences of the theory. By the way, it is well

known that such full understanding is necessary to achieve the categoricity

17

.

By its categoricity, is immediate to show that the compactness Theorem, which

is a corollary of the semantic completeness Theorem, does not hold for FSOA.

This allows the conclusion that the language of FSOA, ie the full second-order

Logic, is not semantically complete. A more general alternative for this conclu-

sion is obtained applying the Lwenheim-Skolem Theorem (L-S ). In a simplied

form that includes both versions (up and down), the L-S Theorem may be stated

in this way: every system equipped with a semantically complete language and

with at least one innite model, admits models of any innite cardinality. Since

models of dierent cardinality cannot be isomorphic, we have that for a system

15

K. Gdel, op. cit., doc. 1929 and 1930, vol. 1, p. 61 and p. 103.

16

L. Henkin, The completeness of the rst-order functional calculus, The journal of symbolic

logic (1949), n.14, pp. 159-166.

17

A full explanation, for example, in S. Shapiro, Foundations without foundationalism: a

case for second-order Logic, Oxford univ. press (2002), p. 82 and p. 112.

7

with innite models, categoricity and semantic completeness of its language are

two complementary properties, impossible to be both satised.

At this point an explanation is necessary, which, surprisingly, I never have

found in any publication. Either by the invalidity of compactness Theorem or

by the L-S Theorem, what one really concludes is precisely the semantic incom-

pleteness of the language of FSOA, ie of the full second-order Logic; and not of

the system. Not every system that is expressed by the full second-order Logic (or,

more generally, by any other semantically incomplete language) has necessarily

to be semantically incomplete

18

. In particular, for a categorical (non-trivial)

system (which always, by the L-S Theorem, uses a semantically incomplete

language), there is nothing to prohibit a priori that it can be syntactically

(or semantically

19

) complete. Indeed, such a system, which necessarily will be

non-formal, could deduce all the true sentences using an appropriate kind of

inherently semantic deduction

20

.

On the other hand, a source of mistakes is undoubtedly the current widespread

tendency to categorize the axiomatic Theories looking at the expression order

(rst order, second order, etc.) without making the necessary distinctions. It is

in line with the disuse of the term formal in its pure Hilbertian sense, used also

by Gdel. Regardless to clarify if the semantics is full (or standard) or general

(ie accomplishing the formality) or intermediate (previous case 3), the systems

of second (or more) order are normally considered as those for which the se-

mantic completeness, and properties related to it, does not apply, as opposed

to those of rst order. But, rstly, a rst order classic system can own proper

axioms that violate either the semantic completeness or the same formality

21

.

Moreover, Henkin has shown that under conditions of formality, the semantic

completeness applies whatever the expression order. As we noted in the previous

section, the crucial aspect lies in the semantic consequences of the cardinality

of language of the theory, regardless of the expression order. Now, the only

peculiarity related to use of the quantiers and on predicative variables (orsuper-predicative, etc.) is that when innite models exist (so, in non-trivial

cases), these variable - if appropriate comprehension axioms are missing - vary

inside uncountable sets, creating inherently semantic systems. But comprehen-

sion axioms can radically change the things. A supercial interpretation of the

Lindstrm's Theorem aggravates the situation. It states that every classical

theory expressed in a semantically complete language can be expressed in a

rst-order language. This can is not a must. The theorem does not prohibit

semantic completeness or even formality for a system expressed in an higher-

18

While, conversely, a system that uses a semantically complete language always is seman-

tically complete.

19

For a categorical system it is immediate to prove that the semantic and syntactic com-

pleteness are equivalent.

20

More in G. Ragun, I conni logici della matematica, ed. Aracne, Roma (2010), pp.

144-145 and 232-233.

21

See, for examples: M. Rossberg, First-Order Logic, Second-Order Logic and Complete-

ness, Hendricks et al. (eds.) Logos Verlag Berlin (2004), on WEB :

http://www.st-andrews.ac.uk/~mr30/papers/RossbergCompleteness.pdf

8

order language

22

. It only states that, when you have this case, the theory can

be re-expressed in a - simpler - rst-order language. Certainly, this property

stands out the particular importance of the rst-order language

23

. But this

should not be radicalized. The grouping of the axiomatic theories based on the

expression order is, in general, misleading their basic logical properties, unless

the actual eect of the axioms is considered. Because these properties are only

a consequence of the premises. The essential tool for classication remains the

accomplishment of the Hilbertian formality.

Starting from the FSOA, the formality can be restored by limiting the in-

duction principle to the formally expressible properties, so re-establishing an

injective correspondence between formulas and properties: in this way the sys-

tem PA is obtained, which however is unable to express, and therefore also

decide, innite (namely, again 2

0, since this number remains unchanged for

subtraction of 0) properties of the natural numbers. Now the induction princi-ple is not longer able to reject all the models not isomorphic to N, so categoricity

cannot be achieved

24

. Indeed, it can be shown in at least four dierent ways

that PA has got non-standard models: 1. by the rst incompleteness Theorem

together with the semantic completeness Theorem; 2. using the compactness

Theorem; 3. by the upward L-S Theorem; 4. as a consequence of a Skolem's

theorem of 1933 that concludes the non-categoricity for PAT

25

.

Finally we recall that the incompleteness Theorems are valid for any eec-

tively axiomatizable (and therefore formal) system in which the general recursive

functions are denable. Therefore, they can be applied to PA but not to FSOA.

For PA, the rst Theorem reveals a further limitation: if one admits its consis-

tency, even between the properties of the natural numbers that this theory can

express, there are innite (0) that it cannot decide.

4 From error to horror

In the third volume of the aforementioned Kurt Gdel's collected works, pub-

lished in 1995, are collected the unpublished writings of the great Austrian

logician. According to the editors, the document *1930c is, in all probability,

the text presented by Gdel at the Knigsberg congress on September 6, 1930

26

.

22

For the second order, you can have a semantically complete language without it being

formal, in one of the cases incorporated in previous point 3.

23

A near property, on the other hand, is already evident thanks to the expressive capacity

of the formal Set Theory: namely, every formal system, since is fully representable in this

theory - which is of the rst order - can be expressed at the rst order.

24

See, for example: S. Shapiro, op. cit., p. 112.

25

T. Skolem, ber die Unmoglichkeit..., Norsk matematisk forenings skrifter, series 2, n. 10,

73-82, reprinted in Selected works in logic, edited by Jens E. Fenstad, Univ. di Oslo (1970),

pp. 345-354.

26

Indeed, in the textual notes of the volume is written (p. 439):

The copy-text for *1930c [...] was one of several items in an envelope that

Gdel labelled Manuskripte Korrekt der 3 Arbeiten in Mo[nats]H[efte] +Wiener

Vortrge ber die ersten zwei (manuscripts, proofs for the three papers in

Monatshefte [1930, 1931, and 1933i ] plus Vienna lectures on the rst two.)

9

In the rst part of the document, Gdel presents his semantic complete-

ness Theorem, extended to the restricted functional calculus (with no doubt

27

identiable with the rst-order classical Logic), which an year before he proved

in his doctoral thesis. After this exposure, he adds:

[...] If the completeness theorem could also be proved for the

higher parts of logic (the extended functional calculus), then it could

be shown in complete generality that syntactical completeness fol-

lows from monomorphicity [categoricity]; and since we know, for

example, that the Peano axiom system is monomorphic [categori-

cal], from that the solvability of every problem of arithmetic and

analysis expressible in Principia mathematica would follow.

Such an extension of the completeness theorem is, however, im-

possible, as I have recently proved [...]. This fact can also be ex-

pressed thus: The Peano axiom system, with the logic of Prin-

cipia mathematica added as superstructure, is not syntactically com-

plete

28

.

In summary, Gdel arms that is impossible to generalize the semantic com-

pleteness Theorem to the extended functional calculus. In fact, in this case also

the Peano axiomatic system, structured with the logic of Principia Mathemat-

ica (PM ), would be semantically complete. But since this theory is categorical,

would follow that it is also syntactically complete. But just this last thing is

false, as he - surprise - announces to have proved.

Now, regardless of what Gdel meant by extended functional calculus, this

armation contains an error. We have in fact two cases:

a. If Gdel understands by Peano axiomatic system structured with the logic of

the PM the formal theory PA (or any other formal arithmetical system)

the error is precisely to regard it as categorical.

b. If, however, he alludes to the only categorical arithmetic, that is FSOA, then

Gdel errs applying to it his rst incompleteness Theorem.

Of the two, just the second belief has been consolidating but - shockingly -

without reporting the error. Rather, exalting the merit of having detected for

the rst time the semantic incompleteness of the full second-order Logic. Prob-

ably, to forming this opinion has been important the inuence of the following

On the basis of that label, *1930c ought to be the text of Gdel's presentation

to Menger's colloquium on 14 May 1930 - the only occasion, aside from the meet-

ing in Knigsberg, on which Gdel is known to have lectured on his dissertation

results [...]. Internal evidence, however, especially the reference on the last page

to the incompleteness discovery, suggests that the text must be that of the later

talk. Since no other lecture text on this topic has been found, it may well be that

Gdel used the same basic text on both occasions, with a few later additions.

27

To be convinced, just consult the note 3 in K. Gdel, op. cit., doc. 1930, vol. 1, p. 103.

28

K. Gdel, op. cit., doc. *1930c, vol. 3, pp. 27-29.

10

sentence contained in the second edition (1938) of Grundzge theoretischen der

Logik by Hilbert and Ackermann

29

:

Let us remark at once that a complete axiom system for the uni-

versally valid formulas of the predicate calculus of second order does

not exist. Rather, as K. Gdel has shown [K. Gdel, ber formal

unentscheidbare Stze der Principia Mathematica und verwandter

systeme, Mh. Math. Physik Vol. 38 (1931)], for any system of prim-

itive formulas and rules of inference we can nd universally valid

formulas which cannot be deduced.

The echo of the Gdel's unfortunate words at the Congress pushes the au-

thors (probably Ackermann, given the age of Hilbert) to attest that the rst

incompleteness Theorem concludes precisely the semantic incompleteness of the

second-order Logic! False; and, furthermore, wrong.

Sadly, today the belief that the incompleteness Theorems even can apply to

FSOA and - above all - that they have as a corollary the semantic incompleteness

of the full second-order Logic is almost unanimous. This can be seen either in

Wikipedia or in the most specialized paper. Even in the introductory note of

the aforementioned document, Goldfarb writes:

Finally, Godel considers categoricity and syntactic completeness

in the setting of higher-order logics. [...] Noting then that Peano

Arithmetic is categorical - where by Peano Arithmetic he means the

second-order formulation - Gdel infers that if higher-order logic is

[semantically] complete, then there will be a syntactically complete

axiom system for Peano Arithmetic. At this point, he announces

his incompleteness theorem: The Peano axiom system, with the

logic of Principia mathematica added as superstructure, is not syn-

tactically complete. He uses the result to conclude that there is no

(semantically) complete axiom system for higher-order logic

30

.

So interpreting, with no doubt, that Gdel refers to the second-order categorical

Arithmetic. But in this case the incompleteness Theorem could not be applied!

Goldfarb neglects that the full second-order induction principle, the only capable

to ensures categoricity, generates an uncountable quantity of axioms, so that the

eective axiomatizability of the system is not veried. As a matter of fact, the

FSOA theory could be syntactically complete.

Often, the semantic incompleteness of the full second-order Logic is con-

cluded by an alternative approach to passing by the (alleged) syntactic incom-

pleteness of FSOA (a Freudian stimulus?). It is supposed, by contradiction, that

the valid statements of the full second-order are eectively enumerable theorems

and then, by applying the incompleteness (or Tarski's) Theorem, an absurdity is

29

English translation by L. M. Hammond et al. in Principles of Mathematical Logic,

Chelsea, New York (1950), p. 130.

30

W. Goldfarb, Note to *1930c, op. cit, vol. 3, pp. 14-15.

11

obtained

31

. But this assumption is evidently excessive: semantic completeness

of a system, simply requires that all the valid statements are theorems, not nec-

essarily eectively enumerable. In the present case, conversely, we already know

that this latter condition is impossible, since, as has been shown, the FSOA is

an inherently semantic theory. As a consequence, these proofs really do not

conclude the semantic incompleteness of the full second-order Logic (as, how-

ever, it can be done by the L-S Theorem or the invalidity of the compactness

Theorem).

The main reasons of this unfortunate misunderstanding are due probably to

ambiguities of the used terminology, both ancient and modern.

5 Clearing up the terms

The expression extended predicate calculus is for the rst time used by Hilbert

in the rst edition (1928) of the aforementioned Grundzge der theoretischen

Logik where, with no doubt, indicates the full second-order Logic. This one was

considered for the rst time in the Principia Mathematica (PM ). The belief that

Gdel, in the aforementioned phrase, refers to FSOA (explanation b.), implies

that he with extended functional calculus intends the same thing. But in which

work he has shown or at least suggested that the incompleteness Theorems can

apply to the full second-order Logic? In none.

In his proof of 1931, Godel refers to a formal system with a language that, in

addition to the rst-order classical logic, allows the use of non-bound functional

variables (ie without the possibility to quantify on them)

32

. Then he proves that

this is not a real extension of the language, able, in particular, to hinder the

applicability of the semantic completeness Theorem. In the 1932b publication,

Gdel declares the validity of the incompleteness Theorems for a formal system

(Z ), based on rst-order logic, with the axioms of Peano and an induction

principle dened by a recursive function. Certainly not a full induction. He

adds:

If we imagine that the system Z is successively enlarged by the

introduction of variables for classes of numbers, classes of classes

of numbers, and so forth, together with the corresponding compre-

hension axioms, we obtain a sequence (continuable into the trans-

nite) of formal systems that satisfy the assumptions mentioned above

[...]

33

speaking explicitly of comprehension axioms and formal systems. Finally, in

the publication of 1934, which contains the last and denitive proof of the rst

31

Emblematic examples, respectively, in: J. Hintikka, On Gdel, Wadsworth Philosophers

Series (2000), p. 22 and S. Shapiro, Do Not Claim Too Much: Second-order Logic and First-

order Logic, Philosophia Mathematica n. 3, vol. 7 (1999), p. 43.

32

K. Gdel, op. cit., vol. 1, p. 187. The purpose is simply to be able to express formulas

such as x(f(x)), where f is a recursive function, which strictly are not permitted in therst-order classical Logic.

33

K. Gdel, op. cit., vol. 1, p. 237.

12

incompleteness Theorem, Gdel, having the aim both to generalize and to sim-

plify the proof, allows the quantication either on the functional or propositional

variables: a declared type of second-order. However, appropriate comprehension

axioms limit at innite countable the number of propositions

34

. Gdel never

misses an opportunity to point out carefully that always is referring to a formal

system and that, consequently, the statements are enumerable:

Dierent formal systems are determined according to how many

of these types of variables are used. We shall restrict ourselves to the

rst two types; that is, we shall use variables of the three sorts p, q,

r,...[propositional variables]; x, y, z,...[natural numbers variables]; f,

g, h, ...[functional variables]. We assume that a denumerably innite

number of each are included among the undened terms (as may be

secured, for example, by the use of letters with numerical subscripts).

[...]

For undened terms (hence the formulas and proofs) are count-

able, and hence a representation of the system by a system of positive

integers can be constructed, as we shall now do

35

.

Therefore, we are in case 1) of the third section: far from full second-order.

Yet, in the introduction to the same paper, Kleene, in summarizing the work of

Gdel, does not avoid commenting ambiguously:

Quantied propositional variables are eliminable in favor of func-

tion quantiers. Thus the whole system is a form of full second-order

arithmetic (now frequently called the system of analysis)

36

.

But he can mean only that the whole system is a formal version (perhaps as

large as possible) of the full second-order Arithmetic. Maybe is exactly this the

extended functional calculus to which Gdel was referring in the examined

words at the Congress? We will discuss it in the next section.

Another source of mistake is probably related to use of the term metamathe-

matics. Although Godel intends it in the modern broad sense that includes any

kind of argument beyond to the coded formal language of Mathematics (and so

with the possibility of using inherently semantic inferences and/or making use

of the concept of truth), in his theorems he employs this term always limiting it

to a formalizable (though often not yet formalized) use deductive and, indeed,

even decidable. In the short paper that anticipates his incompleteness Theo-

rems

37

, for example, Gdel invokes a metamathematics able to decide whether

a formula is an axiom or not :

[...]

34

K. Gdel, On undecidable propositions of formal mathematical systems, op. cit., vol. 1,

p. 353-354.

35

Op. cit., p. 350 and p. 355.

36

S. C. Kleene, op. cit., p. 339.

37

K. Gdel, op. cit., doc. 1930b, vol. 1, p. 143.

13

IV. Theorem I [rst incompleteness Theorem] still holds for all

-consistent extensions of the system S that are obtained by the ad-dition of innitely many axioms, provided the added class of axioms

is decidable, that is, provided for every formula it is metamathemat-

ically decidable whether it is an axiom or not (here again we suppose

that in metamathematics we have at our disposal the logical devices

of PM ).

Theorems I, III [as the IV, but the added axioms are nite], and

IV can be extended also to other formal systems, for example, to the

Zermelo-Fraenkel axiom system of set theory, provided the systems

in question are -consistent.

But in both the subsequent rigorous proofs, he will formalize this process, which

now is called metamathematical, using the recursive functions, so revealing that,

in the words just quoted, he refers to the usual mechanical decidability. By the

same token, even in the theorem that concludes the consistency of the axiom of

choice and of the continuum hypothesis with the other axioms of the formal Set

Theory, he does the same: he uses the metamathematics only as a simplication,

stating explicitly that all the proofs could be formalized and that the general

metamathematical considerations could be left out entirely

38

.

6 An alternative explanation

As noted, Gdel has never put in writing that his proofs of incompleteness may

be applied to the uncountable full second-order Arithmetic and furthermore it

looks absolutely not reasonable to believe that he deems it

39

. In this section,

therefore, we will examine the other possibility, namely the a. of the fourth

section. It, remember, pretends that Godel in 1930 believed, mistakenly, cat-

egorical a kind of formal arithmetic and, in consequence of his incompleteness

Theorems, semantically incomplete its language. Is this reasonable (or more

reasonable than the previous case)?

Certainly not for the system considered by Gdel in his rst proof of 1931: in

fact, the semantic completeness Theorem applies to it, as Gdel himself remarks

in note no. 55 of the publication

40

. Indeed, this is the rst time in which the

existence of non-standard models for a formal arithmetic is proved: why Gdel

does not report it? The topic deserves a brief analysis.

As we have observed in third section, apart from the use of the incomplete-

ness Theorems, the existence of non-standard models for PA can be proved

by the compactness Theorem, the L-S upward one, or a theorem proved by

Skolem in 1933 (cited in note no. 25); actually, this last one resolves that PAT

is non-categorical, what a fortiori applies to PA. The compactness Theorem

is due precisely to Gdel (1930) and descends from his semantic completeness

38

K. Gdel, op. cit., doc. 1940, vol. 2, p. 34.

39

I report a dierent opinion in the book cited in note no. 20 and in the paper [13], because

at that time my knowledge of Kurt Gdel's collected works was insucient.

40

Op. cit., p. 187.

14

Theorem; but in none of his works Gdel ever uses it

41

. Moreover, despite its

fundamental importance for the model theory, nobody - except Maltsev in 1936

and 1941 - uses it before 1945

42

. Not much more fortunate is the story of the L-S

Theorem. The rst proof, by Lwenheim (1915), will be simplied by Skolem in

1920

43

. In both cases, these theorems are downward versions, able to conclude

the non-categoricity of the formal theory of the real numbers and of the formal

Set Theory, but not of PA. However, Skolem and Von Neumann suspect a much

more general validity of the result

44

. It seems that also Tarski was interested to

this argument at that time, probably getting the upward version of the Theorem

in a seminar of 1928

45

. In any case, the argument continues to have low popular-

ity

46

, at least until the generalization of Maltsev in 1936

47

, which, including for

the rst time the upward version, will allow the general conclusion that all the

theories equipped with an innite model and a semantically complete language

are not categorical.

In this context of disinterest for the topic, Gdel not only is no exception,

but his notorious Platonist inclination pushes him to distrust and/or despise any

interpretation that refers to objects foreign to those that he believes existing

independently of the considered theory. Which, in all plausibility, also believes

unique. As a matter of fact, in the introduction of his rst paper on the semantic

completeness, he shows to believe categorical even the rst-order formal theory

of the real numbers

48

.

On this basis, one can surmise the following alternative for the option a.

When he discovers the non-categoricity of the formal arithmetical system where

his original incompleteness Theorems are applied, Gdel is not so glad and

immediately looks for an extension that, though formal, is able to ensure the

categoricity. Probably he believes to have identied it in a formal version of the

full second order Arithmetic: just that one that will be considered in his gener-

alized proof of the rst incompleteness Theorem of 1934

49

, where quantication

41

This is conrmed by Feferman, op. cit., vol. 1, note n. 23, p. 33.

42

This is attested both by Vaught and Fenstad: op. cit., vol. 1, p. 377 and vol. 2, p. 309.

43

The respective works, relative to the rst-order classical Logic instead of the semantic

completeness, are in: J. V. Heijenoort, From Frege to Gdel, Haward University Press (1967),

pp. 232 - 251 and pp. 254 - 263.

44

Here is an especially interesting remark by Von Neumann in 1925, from An axiomatization

of set theory, in From Frege to Gdel, op. cit., p. 412:

[...] no categorical axiomatization of set theory seems to exist at all; [...]

And since there is no axiom system for mathematics, geometry, and so forth

that does not presuppose set theory, there probably cannot be any categorically

axiomatized innite systems at all.

45

This information is due to Maltsev: in his publication of 1936 (cited in a next note), he

claims to have known it by Skolem.

46

It is signicant, for example, that Hilbert did not mention this theme in the seminary of

Hamburg in 1927: The foundations of mathematics, in From Frege to Gdel, op. cit., p. 464.

47

A. Maltsev, Untersuchungen aus dem Gebiete..., Matematicheskii sbornik 1, 323-336;

English translation in: The metamathematics of algebraic systems: collected papers 1936-

1967, Amsterdam (1971).

48

K. Gdel, op. cit., doc. 1929, vol. 1, pp. 61-63.

49

K. Gdel, op. cit., doc. 1934, vol. 1, p. 346.

15

on the functional and propositional variables are allowed, while respecting the

formality. This hypothesis is consistent with the fact that Gdel could admit the

possibility that this language is semantically incomplete, since in both versions

of his semantic completeness Theorem, he does not allow the use of quantiers

on functional variables

50

. Having in mind this generalization (perfectly den-

able as extended functional calculus), it would explain why in the meantime

he communicates the general result without mentioning the discovery of non-

standard models. This comment, just after his famous announcement at the

Congress of Knigsberg on September 6, 1930, is emblematic

51

:

(Assuming the consistency of classical mathematics) one can even

give examples of propositions (and in fact of those of the type of

Goldbach or Fermat) that, while contentually true, are unprovable

in the formal system of classical mathematics. Therefore, if one ad-

joins the negation of such a proposition to the axioms of classical

mathematics, one obtains a consistent system in which a contentu-

ally false proposition is provable.

Admitted the soundness, here he evades to mention standard models, perhaps

because he plans a more general proof than that one currently available, valid

for a formal arithmetic believed categorical. A theory that, as announced in the

questioned armation, uses the extended functional calculus: inevitably, a

semantically incomplete language due to the syntactic incompleteness together

with the (alleged) categoricity.

Anyway, the Skolem's proof of 1933 should have amazed him: not even the

system, semantically and syntactically complete, of the propositions true for

the standard model, is categorical. A rst disturbing evidence that the non-

categoricity covers all formal (non-trivial) systems, regardless of the syntactic

completeness or incompleteness. Gdel, in reviewing the Skolem's paper, only

observes laconically - nally! - that a consequence of this result, that is the non-

categoricity of PA, was already derivable from his incompleteness Theorems

52

.

Later, in any work (not only in the cited generalization of 1934), he always

will ignore the issue of the categoricity. On the other hand, after the questioned

phrase of 1930, he never will return to state that by his incompleteness Theorems

can be derived the semantic incompleteness of some language.

Finally, the Henkin's Theorem of 1949 will prove that in every formal sys-

tem (and so, anywhere the incompleteness Theorems could be applied) there

is semantic completeness of the language and therefore, if innite models exist,

there can not be categoricity.

7 Conclusions

We summarize briey the conclusions of this paper:

50

Op. cit., doc. 1929, 1930 and 1930a, vol. 1, p. 69, p. 121 and p. 125.

51

Op. cit., doc. 1931a, vol. 1, p. 203.

52

Op. cit., doc. 1934c, vol. 1, p. 379.

16

1. The Richard's paradox can be interpreted as a meta-proof that the se-

mantic denitions are not countable, ie that they are conceivably able to

dene each element of a set with cardinality greater than the enumerable

one (so, each real number). Further, the Berry's paradox shows that a -

nite amount of symbols, dierently interpreted, is able to dene an innite

amount of objects.

2. An axiomatic system that allows us to express and study an uncountable

amount of properties can not be formal.

3. The categorical model of the full second-order Arithmetic is an interpre-

tation that, before satisfying the axioms, associates 2

0dierent proposi-

tions to at least one sentence. The result, thus, is a non-formal axiomatic

system.

4. An axiomatic system can be semantically complete despite employing a

semantically incomplete language. In particular, the full second-order

Arithmetic could be syntactically (and therefore also syntactically, being

categorical) complete.

5. The grouping of the axiomatic theories based on the expression order is,

in general, misleading their fundamental logical properties: these are only

a consequence of the premises. The essential tool for classication remains

the accomplishment of the Hilbertian formality.

6. The text of the Gdel's communication at the conference in Knigsberg

on September 6, 1930 (never published by him), contains a mistake. In

the common understanding, not only this error is not reported, but also

is wrongly deduced, by it, that: a) the incompleteness Theorems can also

be applied to the categorical Arithmetic founded on the full second-order

Logic; b) the semantic incompleteness of the full second-order Logic is a

consequence of the incompleteness Theorems (or of the ensuing Tarski's

truth-undenability Theorem).

7. On the basis of the publications of Gdel and of logic, the previous inter-

pretation is untenable.

8. As an alternative interpretation of the manuscript in question, it is possible

that Gdel is referring to the formal arithmetic considered in his proof

of 1934, in which the quantication on the functional and propositional

variables is allowed. If so, in 1930 he believed categorical this theory

and, as a consequence of its syntactic incompleteness, supplied with a

semantically incomplete language. This explanation is consistent with

the fact that both his original semantic completeness Theorems cannot

be applied to this system, due to the said quantication. This view also

explains why, being more and more evident the diculty for the condition

of categoricity, he never will repeat similar armations.

On the other hand, he never corrected the phrase, plausibly since he might

not have worry about correcting an unpublished text.

17

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19